{"paper":{"title":"Testing Simon's congruence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Lukas Fleischer, Manfred Kufleitner","submitted_at":"2018-04-27T12:06:42Z","abstract_excerpt":"Piecewise testable languages are a subclass of the regular languages. There are many equivalent ways of defining them; Simon's congruence $\\sim_k$ is one of the most classical approaches. Two words are $\\sim_k$-equivalent if they have the same set of (scattered) subwords of length at most k. A language L is piecewise testable if there exists some k such that L is a union of $\\sim_k$-classes. For each equivalence class of $\\sim_k$, one can define a canonical representative in shortlex normal form, that is, the minimal word with respect to the lexicographic order among the shortest words in $\\si"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10459","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}